U.S. patent application number 11/458691 was filed with the patent office on 2009-02-05 for constructive method of peak power reduction in multicarrier transmission.
This patent application is currently assigned to Ramot At Tel Aviv University Ltd.. Invention is credited to Simon Litsyn, Alexander Shpunt.
Application Number | 20090034649 11/458691 |
Document ID | / |
Family ID | 37099441 |
Filed Date | 2009-02-05 |
United States Patent
Application |
20090034649 |
Kind Code |
A1 |
Litsyn; Simon ; et
al. |
February 5, 2009 |
Constructive method of peak power reduction in multicarrier
transmission
Abstract
A plurality of bits is transmitted by partitioning the bits
among n subsets; encoding each subset as a respective symbol;
selecting a balancing vector, in accordance with the symbols, from
a set of size 2.sup.p of codewords of length n in {-1,1};
multiplying each symbol by a corresponding element of the balancing
vector; and transmitting the symbols substantially simultaneously.
Preferably, the set of codewords has a strength of at most about 2
ln .left brkt-bot.i.right brkt-bot.. The balancing vector is
selected either deterministically or probabilistically.
Inventors: |
Litsyn; Simon; (Givat
Shmuel, IL) ; Shpunt; Alexander; (Tel Aviv,
IL) |
Correspondence
Address: |
DR. MARK M. FRIEDMAN;C/O BILL POLKINGHORN - DISCOVERY DISPATCH
9003 FLORIN WAY
UPPER MARLBORO
MD
20772
US
|
Assignee: |
Ramot At Tel Aviv University
Ltd.
Tel Aviv
IL
|
Family ID: |
37099441 |
Appl. No.: |
11/458691 |
Filed: |
July 20, 2006 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
60701081 |
Jul 21, 2005 |
|
|
|
Current U.S.
Class: |
375/295 |
Current CPC
Class: |
H04L 27/2615
20130101 |
Class at
Publication: |
375/295 |
International
Class: |
H04L 27/00 20060101
H04L027/00 |
Claims
1. A method of transmitting a plurality of bits, comprising the
steps of: (a) partitioning the bits among n subsets; (b) encoding
each said subset as a respective symbol; (c) selecting a balancing
vector, in accordance with said symbols, from a set of size 2.sup.p
of codewords of length n in {-1,1}, where p<n; (d) multiplying
each said symbol by a corresponding element of said balancing
vector; and (e) transmitting said symbols.
2. The method of claim 1, wherein said transmitting includes
modulating each of n mutually orthogonal subcarriers with a
corresponding said symbol.
3. The method of claim 1, wherein said set of codewords has a
strength of at most about 2.left brkt-bot.ln n.right brkt-bot..
4. The method of claim 1, wherein said balancing vector is said
codeword that minimizes a PMEPR of an envelope of said transmitted
symbols.
5. The method of claim 1, wherein said balancing vector is selected
by picking said codewords randomly from said set and calculating
corresponding PMEPRs of an envelope of said transmitted symbols,
said selected balancing vector being said codeword, from among said
randomly picked codewords, whose corresponding PMEPR is
smallest.
6. The method of claim 5, wherein said codewords are picked
randomly until one said corresponding PMEPR is less than a
predetermined upper bound.
7. The method of claim 5, wherein said codewords are picked
randomly until a number of said randomly picked codewords exceeds a
predetermined upper bound.
8. The method of claim 1, further comprising the step of
transmitting p bits of side information that indicate which said
codeword has been selected.
9. The method of claim 1, wherein said symbols are transmitted
substantially simultaneously.
10. A transmitter for transmitting a plurality of bits using the
method of claim 1.
11. The transmitter of claim 9, wherein said transmitter is an OFDM
transmitter.
12. A communication system comprising: (a) the transmitter of claim
10; and (b) a receiver for receiving the transmitted bits.
13. A transmitter for transmitting a plurality of bits, comprising:
(a) a mechanism for partitioning the bits among n subsets; and (b)
a modulator for: (i) encoding each said subset as a respective
symbol, (ii) selecting a balancing vector, in accordance with said
symbols, from a set of size 2.sup.p of codewords of length n in
{-1,1}, where (iii) multiplying each said symbol by a corresponding
element of said balancing vector, and (iv) modulating each of n
mutually orthogonal subcarriers with a corresponding said
symbol.
14. The transmitter of claim 13, wherein said mechanism includes a
serial-to-parallel buffer.
15. The transmitter of claim 13, further comprising: (c) a
mechanism for converting said modulated orthogonal subcarriers to
time-domain samples; (d) a mechanism for serializing said
time-domain samples; (d) an digital-to-analog converter for
transforming said serialized time-domain samples into an analog
signal.
16. The transmitter of claim 15, wherein said mechanism for
converting said modulated orthogonal subcarriers to time-domain
samples includes a Fourier transform unit.
17. The transmitter of claim 15, wherein said mechanism for
serializing said time-domain samples includes a parallel-to-serial
converter.
18. A communication system comprising: (a) the transmitter of claim
15; (b) a receiver; and (c) a medium for sending said analog signal
to said receiver; wherein said receiver includes: (i) an
analog-to-digital converter for transforming said analog signal
into received time-domain samples; (ii) a mechanism for
parallelizing said received time-domain samples; (iii) a mechanism
for converting said received time-domain samples to n received
orthogonal subcarriers; and (iv) a demodulator for: (A)
demodulating said received orthogonal subcarriers to recover n
corresponding received symbols, (B) multiplying each said received
symbol by a corresponding element of said balancing vector, and (C)
decoding each said received symbol to obtain a corresponding subset
of received bits.
19. The communication system of claim 18, wherein said mechanism
for parallelizing said received orthogonal subcarriers includes a
serial-to-parallel buffer.
20. The communication system of claim 18, wherein said mechanism
for converting said received time-domain samples into aid received
orthogonal subcarriers includes a Fourier transform unit.
Description
[0001] This is a continuation-in-part of U.S. Provisional Patent
Application No. 60/701,081, filed Jul. 21, 2005
FIELD AND BACKGROUND OF THE INVENTION
[0002] The present invention relates to digital communication and,
more particularly, to a method and system for Peak-to-Mean-Envelope
Power Ratio (PMEPR) reduction in multicarrier transmissions such as
Orthogonal Frequency Division Multiplexing (OFDM) systems.
[0003] One approach to the design of a bandwidth-efficient
communication system in the presence of channel distortion is to
subdivide the available channel bandwidth into a plurality of
equal-bandwidth subchannels, with the bandwidth of each subchannel
being sufficiently narrow that lie frequency response
characteristics of the subchannels are nearly ideal. With each of n
subchannels we associate a subcarrier
exp(i2.pi.(f.sub.0+lf.sub.s)t) (1.ltoreq.l.ltoreq.n) where i is the
square root of -1, f.sub.0 is the carrier frequency and it is the
carrier spacing. In OFDM, the symbol rate of each of the
subchannels is set equal to the separation of adjacent subcarriers
so that the subcarriers are orthogonal over the symbol interval,
independent of the relative phase relationships of the
subcarriers.
[0004] The complex envelope of the resulting multicarrier signal
is
m .xi. ( t ) = t = 1 n .xi. l exp ( 2 .pi. ( f 0 + lf s ) t , t
.di-elect cons. [ 0 , f s - 1 ) ##EQU00001##
where .xi.=(.xi..sub.1, . . . , .xi..sub.n) is a complex vector
with entries drawn from a constellation Q of symbols. The
admissible codewords .xi. constitute a code C. Defining
.theta.=2.pi.f.sub.st gives
m .xi. ( .theta. ) = l = 1 n .xi. r exp ( .theta. l ) , .theta.
.di-elect cons. [ 0 , 2 .pi. ) ##EQU00002##
Then
[0005] P M E P R ( .xi. ) = max .theta. .di-elect cons. [ 0 , 2
.pi. ) m .xi. ( .theta. ) 2 E { .xi. 2 } ##EQU00003## P M E P R ( C
) = max .xi. .di-elect cons. C P M E P R ( .xi. )
##EQU00003.2##
[0006] A major problem with multicarrier modulation in general and
with OFDM systems in particular is this PMEPR, the high
peak-to-average power ratio that is inherent in the transmitted
signal. Large signal peaks occur in the transmitted signal when the
signals in the n subchannels add constructively in phase. Such
large signal peaks may saturate the power amplifier at the
transmitter and thus cause intermodulation distortion in the
transmitted signal. Intermodulation distortion can be reduced by
reducing the power in the transmitted signal, so that the power
amplifier always is operated in the linear range; but such a power
reduction results in inefficient operation of the OFDM system.
[0007] Various solutions to this problem have been proposed. For
example, Jones et al., in U.S. Pat. No. 6,307,892, perform bitwise
addition modulo 2 of the codeword vector with a mask vector that is
selected a priori, to be used with all codeword vectors, so as not
to coincide with any of the possible codeword vectors. The method
of Jones et al., and similar methods, are suboptimal in that they
do not take into account the nature of the data actually being
transmitted. For example, Jones et al. select a single mask vector
to be used with all data.
[0008] The closest prior art solution to the present invention is
that of Sharif and Hassibi, "Existence of codes with constant PMEPR
and related design", IEEE Transactions on Signal Processing vol. 52
no. 10 pp. 2836-2846 (October 2004). Given specific data to
transmit, Sharif and Hassibi selectively change the signs of the
symbols .xi..sub.t to minimize PMEPR. Each symbol .xi..sub.t is
multiplied by the corresponding element .epsilon..sub.t of a
balancing vector .epsilon. of length n, all of whose elements are
either +1 or -1.
[0009] Both the Jones et al. patent and the paper by Sharif and
Hassibi are incorporated by reference for all purposes as if fully
set forth herein.
[0010] The solution proposed by Sharif and Hassibi has an
unsatisfactorily large rate loss. For example, their method gives a
zero rate for BPSK modulation and halves the transmission rate if
QPSK modulation is used.
[0011] There is thus a widely recognized need for, and it would be
highly advantageous to have, a method of PMEPR reduction that would
overcome the disadvantages of presently known methods as described
above.
SUMMARY OF THE INVENTION
[0012] The present invention combines the merits of the balancing
method of Sharif and Hassibi with insights from coding theory to
provide PMEPR reduction with a much lower rate loss than in Sharif
and Hassibi. According to the present invention, the balancing
vector is selected only from a specific class of codewords. For
example, using a systematic code of strength 2.left brkt-bot.ln
n.right brkt-bot. as a pool for balancing vectors provides
deterministic or probabilistic reduction of PMEPR with a very
modest rate loss.
[0013] According to the present invention there is provided a
method of transmitting a plurality of bits, including the steps of:
(a) partitioning the bits among 17 subsets; (b) encoding each
subset as a respective symbol; (c) selecting a balancing vector, in
accordance with the symbols, from a set of size 2.sup.p of
codewords of length n in {-1,1}, where p<n; (d) multiplying each
symbol by a corresponding element of the balancing vector; and (e)
transmitting the symbols.
[0014] According to the present invention there is provided a
transmitter for transmitting a plurality of bits, including: (a) a
mechanism for partitioning the bits among n subsets; and (b) a
modulator for: (i) encoding each subset as a respective symbol,
(ii) selecting a balancing vector, in accordance with the symbols,
from a set of size 2.sup.p of codewords of length n in {-1,1},
where p<n, (iii) multiplying each symbol by a corresponding
element of the balancing vector, and (iv) modulating each of n
mutually orthogonal subcarriers with a corresponding the
symbol.
[0015] According to the basic method of the present invention, a
plurality of bits are transmitted by partitioning the bits among n
subsets. Each subset is encoded as a respective symbol. A balancing
vector is selected, in accordance with the symbols, from a set of
size 2.sup.p of codewords of length n in {-1,1}, where p is less
than n. Each symbol is multiplied by a corresponding element of the
balancing vector. Then, the symbols are transmitted, substantially
simultaneously.
[0016] Preferably, transmitting the symbols includes modulating
each of n mutually orthogonal subcarriers with a corresponding
symbol.
[0017] Preferably, the set of codewords, from which the balancing
vector is selected, has a strength of at most about 2.left
brkt-bot.ln n.right brkt-bot..
[0018] In a preferred deterministic embodiment of the method of the
present invention, the balancing vector is the codeword, from the
set of codewords from which the balancing vector is selected, that
minimizes the PMEPR of the envelope of the transmitted symbols. In
other words, from among all the codewords of the set, the selected
codeword is the codeword that, when used as a balancing vector,
gives the smallest PMEPR.
[0019] In a preferred probabilisitc embodiment of the present
invention, the balancing vector is selected by picking the
codewords randomly from the set of codewords and calculating
corresponding PMEPRs of the envelope of the transmitted symbols.
The selected balancing vector is the codeword, from among the
randomly selected codewords, whose corresponding PMEPR is the
smallest. Most preferably, the codeword are picked randomly until a
codeword is picked whose corresponding PMEPR is less than a
predetermined upper bound. Alternatively, the codewords are picked
randomly until the number of codewords that have been so picked
exceeds a predetermined upper bound.
[0020] Preferably, p bits of side information are transmitted to
indicate which codeword has been selected to be used as the
balancing vector.
[0021] Preferably, the symbols are transmitted substantially
simultaneously.
[0022] A basic transmitter of the present invention transmits a
plurality of bits using the basic method of the present invention.
Preferably, the transmitter is an OFDM transmitter. A communication
system of the present invention includes the basic transmitter and
a receiver for receiving the transmitted bits.
[0023] Another basic transmitter of the present invention, for
transmitting a plurality of bits, includes a mechanism, such as a
serial-to-parallel buffer, for partitioning the bits among n
subsets, and a modulator. The modulator encodes each subset as a
respective symbol. The modulator selects a balancing vector, in
accordance with the symbols, from a set of size 2.sup.p of
codewords of length n in {.times.1,1}, where p is less than n. The
modulator multiplies each symbol by a corresponding element of the
balancing vector. Then the modulator modulates each of it mutually
orthogonal subcarriers with a corresponding symbol.
[0024] In addition to the elements of the basic transmitter, a
preferred transmitter of the present invention includes a mechanism
such as a Fourier transform unit for converting the modulated
orthogonal subcarriers to time-domain samples, a mechanism such as
a parallel-to-serial converter for serializing the time-domain
samples, and a digital-to-analog converter for transforming the
serialized time-domain samples into an analog signal.
[0025] A communication system of the present invention includes a
preferred transmitter of the present invention, a receiver, and a
medium for sending the analog signal from the transmitter to the
receiver. The receiver includes an analog-to-digital converter for
transforming the analog signal into received time-domain samples, a
mechanism such as a serial-to-parallel buffer for parallelizing the
received time-domain samples, a mechanism such as a Fourier
transform unit for converting the received time-domain samples to n
received orthogonal subcarriers, and a demodulator. The demodulator
demodulates the received orthogonal subcarriers to recover n
corresponding received symbols, multiplies each received symbol by
a corresponding element of the balancing vector, and decodes each
received symbol to obtain a corresponding subset of received
bits.
BRIEF DESCRIPTION OF THE DRAWINGS
[0026] The invention is herein described, by way of example only,
with reference to the accompanying drawings, wherein:
[0027] FIG. 1 is a high-level schematic block diagram of a system
of the present invention;
[0028] FIG. 2 is a partial high-level block diagram of a
probabilistic transmitter of the present invention;
[0029] FIG. 3 shows plots of simulated PMEPR by the present
invention using dual-BCH code vectors as balancing vectors.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0030] The principles and operation of PMEPR reduction according to
the present invention may be better understood with reference to
the drawings and the accompanying description.
[0031] The Theory Section discusses the use of balancing vectors
selected from a "balancing set" of size 2.sup.p of codewords of
length n in {-1,1} and of strength 2s, where is an integer between
1 and .left brkt-bot.in n.right brkt-bot. (.left brkt-bot.ln
n.right brkt-bot. is the largest integer less than or equal to ln
n). It is shown that if s=[ln n] then the PMEPR reduction of the
present invention is ln n+2.01 ln ln n, which is asymptotically
stronger than the PMEPR reduction achieved by Sharif and Hassibi
and with a much lower rate loss.
[0032] The stronger the balancing set, up to an asymptotic (in n)
limit of 2.left brkt-bot.ln n.right brkt-bot., the better the PMEPR
reduction, but the stronger the balancing set, the larger the
balancing set, so that the improvement in PMEPR reduction that is
obtained by using a stronger balancing set must be balanced against
the larger size of the stronger balancing set. The smallest
codeword set of a given strength is the dual-BCH code of that
strength. Note that for finite n, the optimal balancing set
strength may be greater than 2.left brkt-bot.ln n.right
brkt-bot..
[0033] The Theory Section describes two variants of the method of
the present invention. The first variant is deterministic: the
codeword that minimizes the PMEPR of the data to be transmitted is
found explicitly by testing all the codewords of the balancing set.
The second variant is probabilistic: codewords are picked randomly
from the balancing set and tested until a codeword that gives an
acceptably low PMEPR is found for the data to be transmitted, or
until a maximum allowed number of picked and tested codewords is
reached.
[0034] The deterministic variant is guaranteed to give the best
PMEPR reduction for the given balancing set. The probabilistic
variant is used in case the balancing set is so large that testing
all members of the set is prohibitively expensive in terms of
computational resources. It is shown in the Theory Section that the
probability distribution of PMEPR improves significantly with each
balancing vector that is tried in the probabilistic variant.
[0035] Referring now to the drawings, FIG. 1 is a high-level
schematic block diagram of a system 10 if the present invention.
FIG. 1 is modeled after FIG. 3B of Arnesen, US Patent Application
Publication No. 2003/0026201, which is incorporated by reference
for all purposes as if fully set forth herein. As shown in FIG. 1,
system 10 includes a transmitter 12 and a receiver 14. Transmitter
12 uses the method of the present invention to modulate data to be
transmitted and transmits the modulated data to receiver 14 via a
multi-channel medium 16. Examples of a suitable multi-channel
medium 16 include a wire, a cable, an optical fiber, a coaxial
cable, a waveguide, a radio-frequency propagation path, an optical
propagation path and a twisted pair cable. Receiver 14 receives and
demodulates the data.
[0036] Transmitter 12 includes a serial-to-parallel buffer 18, a
modulator 20, an Inverse Fast Fourier Transform (IFFT) unit 21, a
parallel-to-serial converter 22 and a digital-to-analog converter
24. Serial-to-parallel buffer 18 partitions the input stream of
digital bits to n parallel substreams. Modulator 20 modulates the n
substreams in accordance with the method of the present invention.
Specifically, given n subsets of q bits each to transmit, modulator
20 encodes each subset as a respective symbol .xi..sub.t.sub.1, and
selects an appropriate balancing vector .epsilon. for the symbols
from a set of size 2.sup.p of codewords of length n in {-1,1}.
Modulator 20 multiplies each symbol .xi..sub.t.sub.1 by the
corresponding element .epsilon..sub.t.sub.1 of the balancing vector
.epsilon. and then modulates the corresponding subcarrier
exp(i2.pi.(f.sub.0+lf.sub.s)t) with the resulting product. This
modulation is done in the frequency domain. IFFT unit 21 converts
the frequency-domain subcarriers into time-domain samples.
Parallel-to-serial converter 22 serializes the time-domain samples
and digital-to-analog converter 24 transforms the time domain
samples into an analog signal that is transmitted to receiver 14
via medium 16.
[0037] Transmitter 12 also transmits to receiver 14, via medium 16,
for each set of nq bits, p bits of side information that indicate
which balancing vector from among the possible 2.sup.p balancing
vectors has been used. As discussed in the theory section, the p
side information bits can be encoded in the symbol vector .xi.
along with the transmitted data bits. Alternatively, the p side
information bits can be transmitted out-of-band.
[0038] Receiver 14 includes an analog-to-digital converter 26, a
serial-to-parallel buffer 28, an IFFT unit 29, a demodulator 30 and
a parallel-to-serial converter 32. Analog-to-digital converter 26
transforms the analog signal received from medium 16 into received
time-domain samples. Serial-to-parallel buffer 28 parallelizes the
received time-domain samples. IFFT unit 29 converts the received
time-domain samples into received frequency-domain orthogonal
subcarriers corresponding to the modulated orthogonal subcarriers
that were transformed by IFFT unit 21 to the time domain.
Demodulator 30 demodulates the received orthogonal subcarriers to
recover, n symbols and then uses its knowledge of the balancing
vector .epsilon., obtained from the p bits of side information, to
multiply each recovered symbol by the corresponding element of
.epsilon.. Demodulator 30 then decodes each recovered symbol to
obtain the corresponding subset of q received data bits
Parallel-to-serial converter 32 serializes the n subsets of
received data bits to a set of nq serial output bits.
[0039] Modulator 20, demodulator 30 and IFFT units 21 and 29 may be
implemented in hardware, firmware or software.
Theory Section
1. Introduction
1.2. Discrete and Continuous Maxima
[0040] .theta. was defined above as 2.pi.f.sub.st. Observe that
.theta. varies continuously over [0,2.pi.). We therefore have to
minimize a function of continuous argument. Fortunately, the
following lemma (G. Wunder and H. Boche, "Peak value estimation of
bandlimited signals from their samples, noise enhancement, and a
local characterization of the neighbourhood of an extremum", IEEE
Trans. Signal Processing, vol. 51, 3, pp. 771-780 (2003)) allows to
reduce the problem to minimization over a discrete set of
samples.
[0041] Lemma 1. Let
P ( .theta. ) = l = 1 n .xi. c p .theta. l . ##EQU00004##
Then, for a>1, such that an is integer,
max .theta. .di-elect cons. [ 0.2 .pi. ) P ( .theta. ) .ltoreq. 1
cos n 2 a max i = 1 , 2 , an P ( .theta. i ) , .theta. i = 2 .pi. i
an ##EQU00005##
[0042] In fact this theorem can be improved for a.epsilon.(1,2).
However, in the context of the present invention this interval does
not seem to be of relevance.
1.2. Bounds by Projections
[0043] Let m.sub..xi..sup.R(.theta.) and m.sub..xi..sup.1(9) be the
real and imaginary parts of m.sub..xi.(.theta.) correspondingly.
Then we have
[0044] Lemma 2.
max .theta. .di-elect cons. [ 0 , 2 .pi. ) m .xi. ( .theta. )
.ltoreq. 2 cos .pi. 2 a max i = 1 , 2 , an [ m .xi. R ( .theta. i )
, m .xi. l ( .theta. i ) ] . ##EQU00006##
[0045] Proof. By the previous lemma, we write
max .theta. .di-elect cons. [ 0 , 2 .pi. ) m .xi. ( .theta. ) .rho.
.ltoreq. 1 cos .pi. 2 a max i = 1 , 2 , an m .xi. ( .theta. i )
.ltoreq. 2 cos .pi. 2 a max i = 1 , 2 , , an [ m .xi. R ( .theta. i
) , m .xi. l ( .theta. i ) ] . ##EQU00007##
[0046] Consequently, we have to simultaneously minimize 2an
expressions |m.sub..xi..sup.R(.theta..sub.1)| and
|m.sub..xi..sup.1(.theta..sub.i)|, i=1, 2, . . . , an. In fact,
this approach can be improved. Indeed, instead of projecting on two
axes (real and imaginary) we may pick a larger number, say b, of
evenly distributed lines passing through the origin,
r i ( .PHI. ) = re ( 2 .pi. i b + .PHI. ) p , r .di-elect cons. ( -
.infin. , .infin. ) , i = 1 , , b . ##EQU00008##
[0047] For a complex number c, let c.sup.(i)(.phi.) be its
orthogonal projection on r.sub.i(.phi.),
|c.sup.(i)(.phi.)|=|(c,r.sub.i(.phi.))|.
[0048] We can also write
c ( i ) ( .PHI. ) = Re { c - ( 2 .pi. i b + .PHI. ) p } .
##EQU00009##
[0049] Straightforward analysis similar to the proof of the
previous lemma gives then the following statement.
[0050] Lemma 3. For any complex number c,
c .ltoreq. 1 cos ( .pi. 2 b ) max i = 1 , , b c ( i ) ( .PHI. ) .
##EQU00010##
[0051] If c belongs to the set
R = { c : c = r j 2 .pi. p h j , r j .di-elect cons. R , j
.di-elect cons. Z } , ##EQU00011##
and b divides h, then choosing
.PHI. = .pi. 2 b - .pi. h , ##EQU00012##
we get
c .ltoreq. 1 cos ( .pi. 2 ( 1 b - 2 h ) ) max i = 1 , , b c ( i ) (
.PHI. ) . ##EQU00013##
[0052] Remark 1.4. The reasoning here is similar to that of Wunder
& Boche where however twice as many projections are used.
Notice also that Lemma 2 is a special case of Lemma 3 when b=2 and
.phi.=0. The lemma can be also further improved for b not necessary
dividing n, by optimization in .phi.. Since the gain we obtain is
negligible for n's of practical interest, we omit the easily
reconstructible cumbersome details.
[0053] In what follows this result will be used for simultaneous
minimization of b.gtoreq.2 projections of m.sub..xi.(.theta.) at
each point
.theta. i , .theta. i = 2 .pi. i an . ##EQU00014##
We will show in Subsection 1.7 that we can implement this
minimization by choosing optimal signs for each subcarrier, thus
transforming the problem into that of joint minimization of ban
bounded linear forms.
1.3. Linear Forms
[0054] Definition 5. A linear form L in n variables (x.sub.1, . . .
, x.sub.n) is
L ( x 1 , , x n ) = j = 1 n a j x j , ##EQU00015##
where all a.sub.j are real. If |a.sub.j|.ltoreq.A for j=1, . . . ,
n, and a finite A>0, the form is called bounded by A.
1.4. Strength of Codes
[0055] Definition 6. Let D be a binary code of length n. The
strength t of D is the maximal number such that for any fixed set
of t positions, as we let the codewords vary over D, every possible
t-tuple (out of 2.sup.t possibilities) occurs in these positions
the same number of times,
D 2 ' . ##EQU00016##
[0056] It is known that a code which is dual to a code with the
minimum distance t+1 has strength t. An example of codes with fixed
or slowly growing with the length strength is given by the codes
dual to BCH codes, see e.g. F. J. MacWilliams and N. J. A. Sloane,
The Theory of Error-correcting Codes (Elsevier, 1977). These codes
have length n=2.sup.m-1, the number of information bits ms (i.e.
the number of codewords is 2.sup.ms), and strength 2s. They are
dual to BCH codes of the minimum distance 2s+1. For the length
n=2.sup.m we will exploit duals of the extended BCH codes (with
extra overall parity check bit), thus obtaining codes of length
n=2.sup.m, size 2.sup.ms+1, and strength 2s+1. These codes can be
evidently considered also as being of strength 2s.
[0057] Remark 7. For the sake of completeness let us elaborate on
implementations of dual BCH codes' encoding. Since these codes are
cyclic their codewords can be generated by LSFR having sm
flip-flops. Another simple implementation is based on the following
less-known description. Let F.sub.q be the finite field of size
q=n+1=2.sup.m, having .alpha. as a primitive element. Let F(x) be
the set of polynomials of the form f(x)=f.sub.1x+f.sub.3x.sup.3+ .
. . +f.sub.2s-1x.sup.2s-1 with the coefficients from F.sub.q.
Clearly |F(x)|=2.sup.ms. Let Tr(x),
Tr(x)=x+x.sup.2+x.sup.2.sup.2+x.sup.2.sup.3+ . . .
+x.sup.2.sup.m-1, be the trace function linearly mapping elements
from F.sub.q to F.sub.2. Then the collection of vectors d,
d.epsilon.{.+-.1}.sup.n, with the components
d.sub.i=(-1).sup.Tr(f(a')), i=1, . . . , 2.sup.m-1, and f running
through F(x), constitutes the dual BCH code of strength 2s. For
smaller than 2.sup.m-1 lengths one can use duals of shortened BCH
codes, thus obtaining codes of length 2.sup.m-l-1, size 2.sup.ms,
and strength 2s.
1.5. Rate of Channel Codes
[0058] Definition 8. Let |C| be the number of possible codewords in
a channel code C. The rave of the code C, chosen from a q-ary
constellation, is
R = 1 n log q C . ##EQU00017##
[0059] The irate hit of the code is 1-R.
1.6. Selective Mapping
[0060] The idea of selective mapping is simple: partition all
possible signals to subsets and pick from each subset a
representative with the minimum PMEPR. More formally, let Q.sup.n
be the collection of all vectors of length n with coordinates
belonging to a constellation Q. Assume there exists a partition of
Q.sup.n into M non-intersecting subsets Q.sub.i of equal size
q n M . ##EQU00018##
The information is conveyed by the index of the chosen subset, and
is transmitted by picking one of the vectors belonging to the
corresponding subset. The channel code C consists of the vectors,
one per subset, possessing the minimum PMEPR among the vectors in
the subset. Thus
P M E P R ( C ) .ltoreq. max i = 1 , , M min .xi. .di-elect cons. Q
i P M E P R ( .xi. ) . ##EQU00019##
The rate of the defined code is
1 - 1 n log q M . ##EQU00020##
[0061] There are several simple methods of defining the
partitioning. E.g. let |Q|=q, M=q.sup.r, and g.sub.1, g.sub.2, . .
. , g.sub.M, be invertible mappings from Q.sup.n-r to itself. Given
an information vector v.epsilon.Q.sup.n-r we determine the minimum
PMEPR of the vectors g.sub.1(v), . . . , g.sub.M(v), and transmit
the best vector along with the index of the best transform (side
information). This will clearly be a vector in Q.sup.n. For
instance, one can choose g.sub.1 to be identity, and g.sub.2 to be
a pseudo-random (scrambling) transform.
1.7. Peak Reduction Scheme of Sharif and Hassibi
[0062] Let us briefly review the peak reduction scheme of Sharif
and Hassibi. Since, by Lemma 2,
max .theta. .di-elect cons. [ 0 , 2 .pi. ) m .xi. ( .theta. )
.ltoreq. 1 cos .pi. 2 a max i = 1 , 2 , , an [ m .xi. R ( .theta. i
) ] 2 + max i = 1 , 2 , , an [ m .xi. l ( .theta. i ) ] 2 , .theta.
i = 2 .pi. i an , m .xi. R ( .theta. i ) = l = 1 n Re ( .xi. l z
.theta. , l ) , .theta. i = 2 .pi. i an , i = 1 , 2 , , a n , m
.xi. I ( .theta. i ) = l = 1 n Im ( .xi. l z .theta. , l ) ,
.theta. i = 2 .pi. i an , i = 1 , 2 , , a n , ##EQU00021##
our minimization problem is equivalent to the problem of
minimization of 2an expressions
|m.sub..epsilon..sup.R(.theta..sub.i)| and
|m.sub..epsilon..sup.1(.theta..sub.i)|, i=1, 2, . . . , an.
[0063] This joint minimization problem is tackled by choosing signs
of each subcarrier. Thus we have to minimize 2an bounded linear
forms
min s .di-elect cons. { - 1 , 1 ) n max i = 1 , 2 , , 2 an L i ( s
) , L i ( s ) = l = 1 n a il s l , with ##EQU00022## a il { Re (
.xi. l z .theta. , l ) i = 1 , 2 , , a n , Im ( .xi. l z .theta. ,
l ) i = a n + 1 , a n + 2 , , 2 a n , .theta. i = 2 .pi. i an .
##EQU00022.2##
[0064] Sharif and Hassibi proposed an efficient algorithm for
choosing the optimal signs for each subcarrier, that
deterministically reduces the PMEPR of a code word
4 E max cos 2 ( .pi. / 2 a ) E av ln ( 4 a n ) := c 1 ln ( 4 a n )
, ( 1.1 ) ##EQU00023##
where E.sub.av is the average energy of the constellation.
[0065] Remark 9. Observe that by using Lemma 3 with integer b>2,
the constants in the above expression can be reduced, giving
min b .gtoreq. 2 2 E max cos 2 ( .pi. / 2 b ) cos 2 ( .pi. / 2 a )
E av ln ( 2 b a n ) . ##EQU00024##
[0066] Since the optimal signs have to be conveyed to the receiver,
we need to transmit n bits about the n chosen signs as a side
information. Therefore, the algorithm features the rate loss of
log.sub.q 2. In a preprint published in 2004 and titled "High rate
codes with bounded PMER for BPSK and other symmetric
constellations", Sharif and Hassibi improved the rate efficiency of
the algorithm, at the expense of PMEPR: the number of tones used
for reduction is
n r log q 2 , ##EQU00025##
for PMEPR of rc.sub.1ln 4an, with c.sub.1 as in (1.1).
2. Code Strength and Balancing Linear Forms
[0067] We hereby establish a connection between the strength of
codes over {-1,1}, and their ability to balance linear forms when
code vectors are used as the sign vectors.
[0068] Theorem 2.1. Let D be a code over {-1,1} of length n and
having strength 2s, and m bounded linear forms
L.sub.i(x.sub.1, . . . , x.sub.n), i=1, . . . , m.
[0069] Then
min d .di-elect cons. D max i L i ( d ) .ltoreq. ( ( 2 s ) ! 2 s s
! i = 1 m ( k = 1 n a ik 2 ) s ) 1 / ( 2 s ) . ( 2.1 )
##EQU00026##
[0070] Moreover, for any real .alpha.>1, randomly chosen
codeword d E D, and i=1, 2, . . . , m,
Prob d .di-elect cons. D ( L i ( d . ) .gtoreq. ( .alpha. m ( 2 s )
! 2 s s ! ( k = 1 n a ik 2 ) s ) 1 / ( 2 s ) ) .ltoreq. 1 .alpha. m
. ( 2.2 ) ##EQU00027##
[0071] Proof. Define
.GAMMA. i = d .di-elect cons. D ( L i ( d ) ) 2 s = d .di-elect
cons. D ( j = 1 n a ij d j ) 2 s . ##EQU00028##
[0072] Rewrite the expression for .GAMMA..sub.i:
.GAMMA. i = d .di-elect cons. D j 1 j 2 s j 1 , j 2 s .di-elect
cons. { 1 , , n } m = 1 2 s a ij m d j m = = j _ m = 1 2 s a ij m d
.di-elect cons. D m = 1 2 s d j m = j _ k = 1 n a ik .tau. k ( j _
) d .di-elect cons. D k = 1 n d k .tau. k ( j _ ) ,
##EQU00029##
where the summation is over all vectors j=(j.sub.1, . . . ,
j.sub.2s), and .tau..sub.k(j) is the number of m's, m=1, . . . ,
2s, such that j.sub.m=k.
[0073] For a given j, if there exists a k, such that .tau..sub.k(j)
is odd, then since D is a strength 2s code, we have
.SIGMA..sub.d.epsilon.D.PI..sub.k=1.sup.nd.sub.k.sup..tau..sup.k.sup.(j)=-
0. Otherwise,
d .di-elect cons. D k = 1 n d k .tau. k ( j _ ) = d .di-elect cons.
D 1 = D . ##EQU00030##
[0074] Let J={j: .tau..sub.k(j) is even for all k}. We thus
have
.GAMMA. i = ( j _ .di-elect cons. l k = 1 n a ik .tau. k ( j _ ) )
D , ( 2.3 ) ##EQU00031##
[0075] It is easily shown that
j _ .di-elect cons. l k = 1 n a ik .tau. k ( j _ ) .ltoreq. ( 2 s )
! 2 s s ! ( k = 1 n a ik 2 ) s . ( 2.4 ) ##EQU00032##
[0076] To see this note that
.SIGMA..sub.j.epsilon.J.PI..sub.k=1.sup.na.sub.ik.sup..tau..sup.k.sup.(j)
and (.SIGMA..sub.k=1.sup.na.sub.ik.sup.2).sup.s contain the same
terms, but with different coefficients:
j _ .di-elect cons. l k = 1 n a ik .tau. k ( j _ ) = { s 1 , , s n
} .di-elect cons. { 0 , 1 , 2 , , s ) s 1 + + s n = s ( 2 s ) ! ( 2
s 1 ) ! ( 2 s 2 ) ! ( 2 s i ) ! a i 1 2 s 1 a i 2 2 s 2 a i n 2 s n
.ident. .ident. { s 1 , , s n } .di-elect cons. { 0 , 1 , 2 , , s )
s 1 + + s n = s K s 1 s 2 s n ( 1 ) a i 1 2 s 1 a i 2 2 s 2 a i n 2
s i , ( k = 1 n a ik 2 ) s = { s 1 , , s n } .di-elect cons. { 0 ,
1 , 2 , , s ) s 1 + + s n = s ( s ) ! ( s 1 ) ! ( s 2 ) ! ( 2 s i )
! a i 1 2 s 1 a i 2 2 s 2 a i n 2 s n .ident. .ident. { s 1 , , s n
} .di-elect cons. { 0 , 1 , 2 , , s ) s 1 + + s n = s K s 1 s 2 s n
( 2 ) a i 1 2 s 1 a i 2 2 s 2 a i n 2 s n . ##EQU00033##
[0077] To get (2.4), note
K s 1 s 2 s n ( 1 ) K s 1 s 2 s n ( 2 ) = ( 2 s ! ) s ! s 1 ! ( 2 s
1 ) ! s 2 ! ( 2 s 2 ) ! s i ! ( 2 s i ) ! .ltoreq. ( 2 s ) ! s ! 1
2 s . ##EQU00034##
[0078] Consequently,
.GAMMA. i .ltoreq. D ( 2 s ) ! 2 s s ! ( k = 1 n a ik 2 ) s .
##EQU00035##
[0079] Furthermore,
i = 1 m .GAMMA. i = i = 1 m d .di-elect cons. D ( L i ( d ) ) 2 s d
.di-elect cons. D i = 1 m ( L i ( d ) ) 2 s .ltoreq. D ( 2 s ) ! 2
s s ! i = 1 m ( k = 1 n a ik 2 ) s . ##EQU00036##
[0080] Since all (L.sub.i(d)).sup.2s.gtoreq.0, from the last
inequality it follows that for some d'.epsilon.DA,
i = 1 m ( L i ( d ' ) ) 2 s .ltoreq. ( 2 s ) ! 2 s s ! i = 1 m ( k
= 1 n a ik 2 ) s . ##EQU00037##
[0081] Therefore, for i=1, . . . , m,
L i ( d ' ) .ltoreq. ( ( 2 s ) ! 2 s s ! i = 1 m ( k = 1 n a ik 2 )
s ) 1 / ( 2 s ) , ##EQU00038##
proving (2.1).
[0082] Using the Chebyshev inequality,
Prob d .di-elect cons. D ( L i ( d ) .gtoreq. ( .alpha. m ( 2 s ) !
2 s s ! ( k = 1 n a ik 2 ) s ) 1 / ( 2 s ) ) .ltoreq. E d .di-elect
cons. D { L i 2 s ( d ) } .alpha. m ( 2 s ) ! 2 s s ! ( k = 1 n a
ik 2 ) s , ##EQU00039##
we establish the correctness of (2.2).
[0083] With no further assumptions about the nature of the
coefficients a.sub.ik, we have the following
Corollary 2. Under the conditions of Theorem 2.1 , min d .di-elect
cons. D max i L i ( d ) .ltoreq. ( m ( 2 s ) ! 2 s s ! ) 1 / ( 2 s
) n max i = 1 , 2 , , m j = 1 , 2 , , n a ij , ( 2.5 ) Prob d
.di-elect cons. D ( max i L i ( d ) .gtoreq. ( .alpha. m ( 2 s ) !
2 s s ! ) 1 / ( 2 s ) n max i = 1 , 2 , , m j = 1 , 2 , , n a ij )
.ltoreq. 1 .alpha. , Moreover , for .alpha. ij .ltoreq. 1 , for all
i , j , m = a b n , s = ln n , n .gtoreq. 2 , ( 2.6 ) min d
.di-elect cons. D max i L i ( d ) .ltoreq. 2 n ln n ( a b ) 1 / ( 2
ln n ) ( 1 + 1 / ( 4 ln n ) ) , ( 2.7 ) Prob d .di-elect cons. D (
max i L i ( d ) .gtoreq. 2 .alpha. n ln n ( a b ) 1 / ( 2 ln n ) (
1 + 1 / ( 4 ln n ) ) ) .ltoreq. 1 n ln .alpha. , Proof . To get (
2.5 ) and ( 2.6 ) , put a ik .ltoreq. max i = 1 , 2 , , m j = 1 , 2
, , n a ij in ( 2.1 ) , ( 2.2 ) . For ( 2.7 ) , ( 2.8 ) , use ( 2.8
) 2 .pi. n n n - n .ltoreq. n ! .ltoreq. 2 .pi. n n n - n + 1 12 n
, ln 2 4 ln n + 1 4 D ( ln n ) 2 < 1 + 1 4 ln n , n .gtoreq. 2.
( 2.9 ) ##EQU00040##
3. Method of the Present Invention for PMEPR Reduction
3.1. Strength of Codes and PMEPR Reduction
[0084] Let D be a code from {-1,1}.sup.n of strength 2s. The
vectors d.epsilon.D are candidates for being chosen as the sign
vectors. Let .xi..epsilon.Q.sup.n. Given a--the oversampling
factor, b--the number of projection axes, and .phi.--the projection
angle, we are facing joint minimization of ban bounded linear
forms,
L i ( d ) = L i ( d 1 , , d n ) = l = 1 n a il d l ,
##EQU00041##
where
.theta. i = 2 .pi. i an ##EQU00042##
and
a il = { Re { .xi. l i ( .theta. i l - 2 .pi. b ) } , i = 1 , 2 , ,
an , Re { .xi. l i ( .theta. i l - 4 .pi. b ) } , i = an + 1 , an +
2 , , 2 an Re { .xi. l i ( .theta. i l - 2 .pi. b b ) } , i = ( b -
1 ) an + 1 , ( b - 1 ) an + 2 , , ban } ( 3.1 ) ##EQU00043##
[0085] For the case when the linear forms are given by (3.1), a
more thorough analysis of the structure of (2.1), allows to state
the following bound we give here without proof.
[0086] Theorem 3.1. Under the conditions of Theorem 2.1, with m=ban
linear forms, given by (3.1),
min d .di-elect cons. D max i L i ( d ) .ltoreq. M , ( 3.2 )
##EQU00044##
for .alpha.>1,
Prob d .di-elect cons. D ( max i L i ( d ) .gtoreq. .alpha. M )
.ltoreq. 1 .alpha. s , ( 3.3 ) ##EQU00045##
where for MPSK,
M = ( ( 2 s ) ! 2 s s ! ) 1 / ( 2 s ) ( b a n [ ( n 2 ) s ( 1 + n -
1 - ln 2 2 2 ) s + n - ln 2 n s ] ) 1 / ( 2 s ) ##EQU00046##
[0087] Similar bounds can be derived for other reflection-symmetric
constellations, e.g. QAM. We omit the cumbersome details.
[0088] Corollary 3.2. Under the conditions of Theorem 3.1, for s=ln
n, n.gtoreq.2,
min d .di-elect cons. D max i L i ( d ) .ltoreq. n ln n ( 1 + 3 ln
n ) , Prob d .di-elect cons. D ( max i L i ( d ) .gtoreq. .alpha. n
ln n ( 1 + 3 2 ln n ) ) .ltoreq. 1 n ln .alpha. . ##EQU00047##
[0089] Proof.
Use ( ( 1 + n - 1 - ln 2 2 2 ) ln n + 1 ) 1 / ( 2 ln n ) < 1 + 1
ln n , together with ( 2.9 . ##EQU00048##
[0090] Theorem 3.3. Let D be a code of strength 2s from
{-1,1}.sup.n. For every .xi..epsilon.Q.sup.n, there exists a
d.epsilon.D, such that
PMEPR ( .xi. * d ) .ltoreq. := = E max E av min a > 1 , a n
.di-elect cons. N min b > 1 , b .di-elect cons. Z { a 1 / s cos
2 .pi. / 2 a b 1 / s cos 2 .pi. / 2 b ( ( 2 s ) ! n 2 2 s s ! ) 1 /
s ( 1 + 2 / s ) } , .xi. * d := ( .xi. 1 d 1 , , .xi. n d n ) . (
3.4 ) ##EQU00049##
[0091] Since Q is reflection-symmetric, .xi.*d.epsilon.Q.sup.n.
[0092] Proof. Use Theorem 2.1 combined with Lemma 1.3 and the
definition of PMEPR. Also use the inequality ((1+1/ {square root
over (2n.sup.1-ln 2)}).sup.s+1).sup.1/s<1+2/s, n.gtoreq.2.
[0093] Corollary 3.4. Under the conditions of Theorem 3.3, for s=ln
n, and for all n.gtoreq.n.sub.0, we have
.ltoreq. n ln n E max E av ( 1 + .sigma. n 0 ln ln n ln n ) ,
##EQU00050##
[0094] where .sigma..sub.64=22, .sigma..sub.128=17,
.sigma..sub.2048=9, and .sigma..sub.n.sub.0=1+.epsilon.,
.epsilon.>0 becoming arbitrary small for large n.sub.0.
[0095] Proof. Choose a=b= {square root over (ln n)}, and use
standard inequalities.
3.2. PMEPR Reduction Scheme
[0096] Let D be a code of strength 2s from {-1,1}.sup.n of size
2.sup.p. The following particular case of selective mapping is
used. Let .xi.=.xi.=(.xi..sub.1, . . . ,
.xi..sub.n).epsilon.Q.sup.n be the vector we wish to transmit.
Compare PMEPR of 2.sup.p vectors, .xi.*d, where d runs over D, and
send the signal corresponding to .xi.*=.xi.*d' with the minimum
PMEPR, along with the side information of p bits indicating which
balancing vector has been chosen. This allows at the receiver to
recover d' by encoding the p information bits into the
corresponding word from D, and therefore reconstruct the vector:
.xi.=.xi..sup.**d'. We arrive at the following result.
[0097] Theorem 3.5. Let D be a binary linear systematic code of
strength 2s and size 2.sup.p. Then there exists a scheme for PMEPR
reduction guaranteeing PMEPR not exceeding Y from (3.4) with the
rate hit
p log q 2 n ##EQU00051##
and complexity proportional to n2.sup.p.
[0098] Using duals of BCH codes we obtain the following
corollary.
[0099] Corollary 3.6. The PMEPR reduction scheme of the present
invention guarantees the maximum PMEPR of Y defined in (3.4) with
the rate hit
s log q ( n + 1 ) n . ##EQU00052##
[0100] Remark 3.7. Notice that to compute PMEPR in die algorithm it
is necessary to calculate the values of an complex linear forms,
the projection on axes is used only in the proof. Starting from
Theorem 3.3 we neither have used minimization in the starting
projection angle .phi., nor have we taken into account that the
factor A can be simultaneously strictly less than 1 for all the
forms. Clearly this can be used in computations for particular
cases. However, we did not find examples where this provides a
significant difference.
[0101] Remark 3.8. Transmission of the side information is an
important issue in implementation of the algorithm of the present
invention. In what follows we discuss several options. We assume
that the signal .xi. is obtained as a result of coding which can be
distorted by the following multiplication by a balancing vector. A
choice at the receiver is that we may either first multiply by the
balancing vector followed by decoding, or start from decoding and
then multiply by the balancing vector.
[0102] The simplest situation is when there exist very reliable
uncoded bits which can be used for conveying the index of the
balancing vector (e.g. when only one or two bits from constellation
of size 8 or more are protected by error-correcting code). If these
bits are mapped to antipodal constellation points this does not
affect the resulting PMEPR.
[0103] Another possibility is that we have p reliable subcarriers
(this can be achieved e.g. by decreasing the size of the
constellation in these subcarriers). Without loss of generality
assume that these p subcarriers are the first ones, otherwise a
permutation of the balancing vectors should be used. Let D be a
systematic code, i.e. having the information bits at its first p
positions. Let Q* be a half of the constellation Q, in which we
pick one out of every pair of antipodal points. Let
.xi.=(.xi..sub.1, . . . , .xi..sub.n) with .xi..sub.1, . . . ,
.xi..sub.p.epsilon.Q*, and .xi..sub.p+1, . . . ,
.xi..sub.n.epsilon.Q. Compare PMEPR of 2.sup.p vectors, .xi.*d,
where d ruins over D, and send the signal corresponding to
.xi.*=.xi.*d' with the minimum PMEPR. At the receiving end one
deduces the binary information vector of d' by checking if in the
received vector .xi.* each of the first p components belongs or
does not belong to Q*.
[0104] In this setting it is also possible to compress the
information about the chosen code vector to the nearest integer
greater than s log.sub.q(n+1) tones (perhaps reserved). This allows
further minimization of the number of the subcarriers affected by
the algorithm. In this case the minimization of PMEPR is done for
the signal vector containing the transmitted information. This
however yields a slight increase by s log.sub.q(n+1) in the
estimate for PMEPR.
[0105] Now consider the situation when we prefer to decode first
and only then to subtract the balancing vector. Let the transmitted
information be protected by some error-correcting code D', i.e.
only vectors .xi..epsilon.D'.OR right.Q.sup.n are sent. To start
from decoding in D' we have to guarantee that the modified vector
always belongs to D'. For instance, if q=2, i.e. when we use BPSK,
and D' is a linear code, it is sufficient that the code D we use
for balancing is a subcode of D'. Then the modified vector .xi.*
also belongs to D' and can be decoded without knowledge of the
balancing vector. For higher than BPSK constellations and use of
linear code, the embedding D.OR right.D' provides a sufficient
condition for this scheme to work. This embedding is not very
restrictive. For example, if D is a dual BCH code of fixed
strength, it is possible to show that it is nested in BCH codes
with a constant minimum distance. As well it is possible to design
efficient LDPC codes containing duals of BCH codes. We plan to
address the problem of constructing such codes elsewhere.
4. Probabilistic Analysis and a Practical Scheme
[0106] In the previous sections we provided deterministic and
probabilistic bounds on PMEPR, using balancing vectors from codes
of given strength. In this section we will demonstrate how a
practical scheme can be designed based on the above, and provide a
probabilistic analysis of such scheme.
[0107] Indeed, implementation of the full deterministic scheme for
meaningful s is computationally challenging. However, by picking at
random at most a fixed number of balancing vectors from the code,
we could guarantee achieving arbitrary close to 1 probability of
PMEPR restricted to the derived deterministic bound. Following the
Remark 3.8, the possible implementations of the scheme vary
accordingly with the chosen method of the balancing vector
transmission. One possible implementation of the scheme is
presented in the block-scheme on FIG. 2. Other implementations are
possible, differing in the stop criterion (when the vector is
transmitted: after a fixed number of candidates tried, or when the
resulting PMEPR is small enough), as well as other details.
[0108] To analyze the probabilistic scheme of the present
invention, assume that s=ln n, and the number of balancing vectors
used is h. Using Chemoff bound (N. Alon and J. Spencer, The
Probabilistic Method (Wiley, 2000)) for real .alpha.>1, and
large n, and its tightness for a single linear form, for a random
channel code C, we have
0.5n.sup.-.alpha..ltoreq.Prob(PMEPR(C).gtoreq..alpha. ln
n).ltoreq.2n.sup.-.alpha.+1 (4.1)
i.e. a polynomial in n decrease.
[0109] Considering another range of PMEPR, we have for
.beta.>0,
Prob(PMEPR(C).gtoreq.ln n+.beta.ln ln n).ltoreq.2 ln.sup.-.beta.n.
(4.2)
[0110] The constants in these expressions can be improved.
[0111] Theorem 4.1. For any .xi..epsilon.Q.sup.n, let d.sub.1,
d.sub.2, . . . , d.sub.h be randomly picked from a code D of
strength 2 ln n. Then, for all n.gtoreq.n.sub.0,
Prob ( min i = 1 , 2 , , h PMEPR ( .xi. * d i ) .gtoreq. .alpha. ln
n + .sigma. n 0 ln ln n ) .ltoreq. n - h ln .alpha. , ( 4.3 ) Prob
( min i = 1 , 2 , , h PMEPR ( .xi. * d i ) .gtoreq. ln n + ( .beta.
+ .sigma. n 0 ) ln ln n ) .ltoreq. n - h ( ln ( 1 + .beta. ln ln n
ln n ) ) , ( 4.4 ) ##EQU00053##
where the constant .sigma..sub.n.sub.0 is given by Corollary
3.4.
[0112] Proof. Immediate from Corollary 3.4.
[0113] Remark 4.2. Indeed, we see that the proposed scheme allows
us to considerably improve the PMEPR statistics, using only (ln
2)(log.sub.2 n)+1 bits of redundancy (the nearest integer greater
or equal to
( ln 2 ) ( log 2 n ) 2 + 1 log 2 q ##EQU00054##
redundant subcarriers), and a modest increase in complexity.
Moreover, our result is mathematically rigorous, applicable to any
reflection-symmetric constellation, and provides the reduction for
any information vector. In other words, for any information vector,
choosing h big enough, we can provably make the probability of the
large PMEPR arbitrary small, up to the deterministic bounds,
attained at h being equal the code size.
[0114] As an example, setting h=.rho.n/(ln n ln .alpha.), we have
for all n.gtoreq.n.sub.0,
Prob ( min i = 1 , 2 , , h PMEPR ( .xi. * d i ) .gtoreq. .alpha. ln
n + .sigma. n 0 ln ln n ) .ltoreq. - pn , ##EQU00055##
[0115] As another example, setting h=n/(ln ln n), for all
n.gtoreq.n.sub.0,
Prob ( min i = 1 , 2 , , h PMEPR ( .xi. * d i ) .gtoreq. ln n + (
.beta. + .sigma. n 0 ) ln ln n ) .ltoreq. - .beta. n ( 1 - .beta.
ln ln n ln n ) , ##EQU00056##
i.e. comparing with (4.1), we transform the polynomially in n
decreasing probability into an exponentially decreasing one.
5. Examples and Simulations
[0116] In the above, we have provided a probabilistic framework for
PMEPR reduction towards certain values, depending on the scheme
parameters (e.g. balancing code strength, oversampling a, number of
axes b). Assume for simplicity that the constellation used has
E.sub.max=E.sub.av (for instance MPSK).
[0117] For any information vector length n, the balancing code
strength 2s prescribes the optimal a, b. Denote the PMEPR bound,
guaranteed for the channel code C, using the balancing code
D.sub.2s of strength 2s (either deterministically, using the whole
code, or probabilistically, using a chosen number of candidates,
for the wanted peak probability reduction), by PMEPR.sub.D.sub.2,
(C). For the balancing code of the least meaningful strength, 2s=4,
say the dual of the extended BCH code of strength 4 (it is dual to
the extended 2-error correcting BCH code), length 2.sup.m, having
2.sup.m+1 words, we need 2m+1 bits to indicate which specific code
word is used. Choosing e.g. a=3, b=3, we obtain
PMEPR.sub.D.sub.s(C).ltoreq.8 {square root over (n/3)}. Using the
optimal strength balancing code, e.g. the dual BCH code of length
n=2.sup.m and strength 2 ln n, having 2.sup.m ln n+1 words, we need
m ln n+1 bits to indicate which specific codeword was used.
[0118] Tables 1 and 2 show PMEPR.sub.D.sub.2, (C) for various
values of parameters for MPSK, n=128 (Table 1) and n=2048 (Table
2). Rate hit for QPSK is also calculated. For simplicity we
restricted ourselves to integer a=b.
TABLE-US-00001 TABLE 1 PMEPR achievable by present invention, MPSK,
n = 128, max. PMEPR = 21.07 dB code's half- 3 4 5 7 9 strength s
optim. const. 4 5 5 6 7 B, a PMEPR.sub.D2s 32.27 21.04 16.84 14.09
13.54 (C) PMEPR.sub.D2s 15.09 13.23 12.26 11.49 11.32 (C) [dB] rate
hit for 0.08594 0.11328 0.14063 0.19531 0.25000 QPSK
TABLE-US-00002 TABLE 2 PMEPR achievable by present invention, MPSK,
n = 2048, max. PMEPR = 33.11 dB code's half-strength s 2 3 4 11
optim. const. B, a 3 4 5 7 PMEPR.sub.D.sub.2s (C) 329.65 76.46
39.40 16.21 PMEPR.sub.D.sub.2s (C) [dB] 25.18 18.83 15.95 12.10
rate hit for QPSK 0.00562 0.00830 0.01099 0.02979
[0119] FIG. 2 shows simulated 10 million runs for QPSK, n=128, with
oversampling a=5, using balancing vectors (BV) randomly chosen from
strength 2s=10 dual-BCH code (only 18 redundant carriers). For
example, the peaks higher than 10.8 dB occur with probability
10.sup.-2. Using 4 BVs, the probability of such peaks is lowered to
10.sup.-5. Looking at it differently, to build a system, for any
peak probability less than 10.sup.-25, we need an amplifier with
the dynamic range reduced by 2 dB, at a modest cost of trying 4
BVs. The complexity can thus be traded for PMEPR reduction, up to
the theoretical limits provided in the previous sections.
[0120] While the invention has been described with respect to a
limited number of embodiments, it will be appreciated that many
variations, modifications and other applications of the invention
may be made.
* * * * *